List of top Linear Algebra Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), not all zero, be such that \(\sum^9_{i=1}a_iv_i=0\). Then the system \(Ax=\sum^9_{i=1}v_i\) has

Let G be a finite group. Then G is necessarily a cyclic group if the order of G is

Which of the following is a subspace of the real vector space \(\R^3\) ?

Let S and T be non-empty subsets of \(\R^2\), and W be a non-zero proper subspace of \(\R^2\). Consider the following statements :
I. If span(S) = \(\R^2\) , then span(S ∩ W) = W.
II. span(S ∪ T) = span(S) ∪ span(T).
Then

Let A ⊆ \(\Z\) with 0 ∈ A. For r, s ∈ \(\Z\), define
rA = {ra : a ∈ A},   rA + sA = {ra + sb : a, b ∈ A}.
Which of the following conditions imply that A is a subgroup of the additive group \(\Z\) ?

Let W be the subspace of \(M_3(\R)\) consisting of all matrices with the property that the sum of the entries in each row is zero and the sum of the entries in each column is zero. Then the dimension of W is equal to ___________.

For σ ∈ S₈, let o(σ) denote the order of σ. Then max{o(σ) : σ ∈ S₈} is equal to __________.

For g ∈ \(\Z\), let \(\bar{g}\) ∈ \(\Z_8\) denote the residue class of g modulo 8. Consider the group \(\Z^×_8\) = {\(\bar{x}\) ∈ \(\Z_8\) : 1 ≤ x ≤ 7, gcd(x, 8) = 1} with respect to multiplication modulo 8. The number of group isomorphisms from \(\Z^×_8\) onto itself is equal to ________

Which of the following is/are true ?

The maximum number of linearly independent eigenvectors of the matrix
\(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3 \\ \end{pmatrix}\)
is equal to _________.

Which one of the following groups has elements of order 1, 2, 3, 4, 5 but does not have an element of order greater than or equal to 6 ?

Let V be a nonzero subspace of the complex vector space 𝑀₇(ℂ) such that every nonzero matrix in 𝑉 is invertible. Then, the dimension of V over ℂ is

For a positive integer n, let U(n) = {\(\bar{r}\) ∈ ℤ_n ∶ gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE ?

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V ∖ W and v ∈ V ∖ W, then u + v ∈ V ∖ W.
Which one of the following holds ?

Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE ?

Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE ?

Let \( V \) be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 6, together with the zero polynomial. Then which one of the following is true?

Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?

Let \( P \in \mathbb{M}_4 (\mathbb{R}) \) be such that \( P^4 \) is the zero matrix, but \( P^3 \) is a nonzero matrix. Then which one of the following is FALSE?

For \( X, Y \in M_2(\mathbb{R}) \), define \((X, Y) = XY - YX\). Let \( 0 \in M_2(\mathbb{R}) \) denote the zero matrix. Consider the two statements:
\[ P : (X, (Y, Z)) + (Y, (Z, X)) + (Z, (X, Y)) = 0 \text{ for all } X, Y, Z \in M_2(\mathbb{R}).\]\[ Q : (X, (Y, Z)) = ((X, Y), Z) \text{ for all } X, Y, Z \in M_2(\mathbb{R}).\]Then which one of the following is correct?

For \( P \in M_5(\mathbb{R}) \) and \( i,j \in \{1,2, \ldots, 5\} \), let \( p_{ij} \) denote the \((i,j)\)th entry of \( P \). Let\[ S = \{ P \in M_5(\mathbb{R}) : p_{ij} = p_{sr} \text{ for } i,j,s,r \in \{1,2, \ldots, 5\} \text{ with } i + r = j + s \}.\]Then which one of the following is FALSE?

For a 4 x 4 matrix \(M\isin M_4(\mathbb{C})\), let M denote the matrix obtained from M by replacing each entry of M by its complex conjugate. Consider the real vector space
H = \(\{M\isin M_4(\mathbb{C}):M^T=\overline M\)
where M^T denotes the transpose of M. The dimension of H as a vector space over \(\R\) is equal to

Let \( G \) be a group of order 2022. Let \( H \) and \( K \) be the subgroups of \( G \) of order 337 and 674, respectively. If \( H \cup K \) is also a subgroup of \( G \), then which one of the following is FALSE?

Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?

For \(g\isin\Z\), let \(\bar{g}\isin\Z_{37}\) denote the residue class of g modulo 37. Consider the group U₃₇ = {\(\bar{g}\isin\Z_{37}:1\le g\le37\) with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?